where \(C(x,t)\)  is the distributions of the particle concentration in space and time and \(D\)  is the diffusion
coefficient. \(C(x,t)\)  in the equation above is considered up to a constant, consequently it may also refer to the
concentration above or around the average. The
function \(C(x,t)\)  fulfills the necessary smoothness conditions with existing continuous first and second derivatives
in respect to time and and space and from physical reasons \(D > 0\) .
Numerous physics textbooks gives us the derivation how the fundamental (the Gaussian) solutions can be obtained e.g. [1]}, [2]}, [3]}, [4]}.
First, to dispel misunderstandings we have to express one thing clearly, the
regular diffusion equation has existence and unicity theorem for initial and boundary problems, but this is not contradictory to our forthcoming analysis. We will apply three
different trial functions (or Ansätze [this is the plural form] ) but neither the initial nor the boundary problems
are being well defined. The obtained results may fulfill well-defined initial and boundary problems via fixing their integration constants \(c_1\)  and \(c_2\) .
